THE ROLE OF FLUENCY IN MATHEMATICAL DEVELOPMENT: FACTORS ASSOCIATED WITH EARLY LEARNING DIFFICULTIES IN MATHEMATICS

Finnane, Maureen Kathryn (2007). THE ROLE OF FLUENCY IN MATHEMATICAL DEVELOPMENT: FACTORS ASSOCIATED WITH EARLY LEARNING DIFFICULTIES IN MATHEMATICSPhD Thesis, School of Education , University of Queensland.

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This thesis explores factors associated with the development of early learning difficulties in mathematics from two perspectives: issues relating to mathematics development and early mathematics education, and models of memory and information processing. A principal aim of the research was to better understand the puzzling question of why a significant number of students with otherwise good reasoning capacity are unable to develop efficient, effective strategies for solving basic number combinations. The very poor mathematical understanding and skills of students who have been described as mathematically disabled (MD) or as having an arithmetic learning difficulty (ALD) or mathematical learning difficulty (MD) have been described in the research literature over a period of thirty years. The profile of the characteristics of MD students is well established, having been described with a consistent prevalence of 6-8% students in several countries. In spite of having average conceptual understanding in some areas of mathematics, these students face an outstanding difficulty in mastering basic arithmetic facts (Ginsburg, 1997), and continue to use very slow and ineffective strategies for solving even basic arithmetic problems, usually reliant on counting-by-ones finger strategies. More recent accounts of MD students also emphasize their poorly developed number sense and estimation skills. The development of effective interventions for mathematical learning difficulties is complicated by the fact that there are no distinctive diagnostic measures available for identifying MD (Geary, 2004). Furthermore, there are now strong indications that there are at least two distinctive forms of MD, with different instructional needs (Jordan, Hanich & Kaplan, 2003; Fuchs, Fuchs & Prentice, 2004). Accordingly, one goal of this thesis was to identify significant predictors of early mathematical learning difficulties, as a guide to identification and intervention. Based on previous research exploring factors associated with MD, a comprehensive range of assessment measures was administered to 68 children in three Year 2 classes in metropolitan Brisbane, Queensland. The measures included detailed assessments of mathematical skills and understanding, and a range of memory and processing tests, to capture underlying factors constraining early mathematical learning. Children at risk of early mathematical difficulties were identified by a statewide process of diagnosis in Queensland administered in the second year of formal schooling, known as the Year 2 Diagnostic Net (Numeracy). In view of the high rate of comorbidity of mathematical and literacy learning difficulties, data were also collected about which students failed the Year 2 Diagnostic Net (Literacy). For the same reason, the childrens speed of naming symbols, letters and words was measured at the same time as their fluency in identifying numbers. To understand better the nature of their learning difficulties, diagnostic teaching sessions were undertaken with ten of the students caught in the Year 2 Net (Numeracy), whom their teachers perceived as most in need of intensive learning support. In addition, an assessment and intervention were conducted with a Year 4 student at one of the study schools, who was facing significant mathematical and literacy learning difficulties. The study showed that forwards counting fluency and stage of strategy development (SEAL) on the Learning Framework in Number (Wright, Martland & Stanger, 2000) in the beginning of Year 2 were significant predictors of being caught in the Year 2 Net (Numeracy) at mid-year. Net status was also predicted by a students performance on the Make 10 test, a novel task designed by the researcher to assess fluency in retrieval of Ten Fact combinations. As well as the level of strategy use (SEAL), a students level of identification of written numerals was a further significant predictor of performance at the end of Year 2 on I Can Do Maths (Doig & de Lemos, 2000), a standardized test of early knowledge of Number, Measurement and Space concepts. The importance of identification and intervention for students at risk of MD was indicated by the increasing gap in performance between Net and normally achieving students on I Can Do Maths by the end of Year 2. The speed of processing of particular numerical skills were shown to be important indicators of early mathematical learning. Net students were significantly slower than their normally achieving peers in identifying 2-digit numbers on the CAAS 2-Digit task, and were slower on all the PAL rapid automatised naming RAN tasks - RAN Digits, Letters, Words, Words & Digits (Berninger, 2001). Fluency in rapidly naming and switching between 2-digit numbers and familiar words, as measured on the RAN Words & Digits task was also shown to be a significant predictor of performance on the One Minute Basic Number Facts Test of Addition. The level of Highest Forward Digit Span and poor performance on a working memory Counting Span task significantly discriminated between Net and normally achieving students. Information processing models and models of working memory provide useful explanatory frameworks to account for particular error patterns, and for the characteristic failure of MD students to learn basic arithmetic facts. Insights from the mathematics development and mathematics education literature add to our ability to understand the early impasse in learning of students at risk of MD. Work samples from diagnostic teaching sessions with five of the Intensive Net students further illustrate the influence of a low stage of counting fluency and strategy use on a students self-perception and ability to access the Year 2 mathematics curriculum. Processing strengths and weaknesses are discussed in relation to possible effects on the students mathematical learning. The detailed case study of the Year 4 student demonstrates how a researchbased intervention, which took into account cognitive factors, was successful in enabling learning of arithmetic facts and markedly changing the motivation and confidence of a student who was formerly very resistant to mathematics instruction. In particular, increased counting fluency and automatisation of addition facts appeared to free working memory resources for monitoring and self-correction by the student, and to facilitate a positive interest in exploring mathematical relationships. Based on the significant predictors of poor achievement in early mathematics, assessment measures are proposed which are critical for the identification of early mathematical learning difficulties. The thesis concludes with a number of recommendations for intervention, with accompanying requirements for pre-service and in-service training to assist effective implementation. Future directions for research in the identification and intervention of MD are proposed.